Study on the arithmetic discontinuous groups

本文的目的之一是明确地确定所有具有签名（O; e_1，e_2，e_3，e_4）的算术Fuchsian群Γ。（1）设K是n次全真实的代数数域K。设A是K上的四元数代数，使得A[叉积]_Q R &lt;$M_2（R）[对称性]H^<n-1>，其中H是汉密尔顿四元数代数.设O是A的一个阶。设Γ^1（A，O）是由O的范数为1的单位群O^1导出的Fuchsian群.若Fuchsian群Γ与Γ^1（A，O）可交换，则称Γ为K-算术Fuchsian群。若存在一个具有签名（O; e_1，e_2，e_3，e_4）的K-算术Fuchsian群Γ，则[K：Q][小于或等于]10.并给出了这类域K的判别式d（K）的显式上界D_0。我们确定了具有最小判别式d（K）的10次非本原域K。（c. f. K. Takeuchi [1]）（2）研究了K=Q的情形。设A是Q上的一个四元数代数，O（f）是A中的一个Eichler序，其水平f是无平方的.设Γ^*（A，O（f））是Γ^1（A，O（f））在SL_2（R）中的正规化子.证明了若Γ是一个Q-算术Fuchsian群.则Γ是Γ^*（A，O）的有限指数子群，（参见K. Takeuchi [2]）从而我们可以明确地确定所有具有签名（O; e_1，e_2，e_3，e_4）的Q-算术Fuchsian群Γ。

One of the aims of this project is to determine all arithmetic Fuchsian groups Γ with signature (O;e_1,e_2,e_3,e_4) explicitely.(1)Let K be a totally real algbraic number field K of dgree n. Let A be a quaternion algebra over K such that A【cross product】_Q R〓M_2(R) 【symmetry】H^<n-1>, where H is the Hamilton quaternion algebra. Let O is an order of A. Let Γ^1(A,O) be a Fuchsian group derived from unit group O^1 of O of norm 1. If a Fuchsian group Γ is commensurable with Γ^1 (A,O), Γ is called K-arithmetic Fuchsian group. If there exists a K-arithmetic Fuchsian group Γ with signature (O;e_1,e_2,e_3,e_4), then the degree [K:Q]【less than or equal】10. Moreover, the explicit upper bound D_0 of the discriminant d(K) of such fields K is given. We have determined the imprimitive field K of degree 10 with minimum discriminant d(K). (c.f.K.Takeuchi [1])(2)We have studied the case K=Q. Let A be a quaternion algebra over Q and let O(f) be an Eichler order in A with square-free level f. Let Γ^*(A,O(f)) be the normalizer ofΓ^1(A,O(f)) in SL_2(R). We show that if Γ is a Q-arithmetic Fuchsian group. Then Γ is a subgroup of Γ^*(A,O) of finite index, (c.f.K.Takeuchi [2])Consequently, we can deternime all Q-arithmetic Fuchsian groups Γ with signature (O;e_1,e_2,e_3,e_4) explicitly.